CSDC Seminar: Shrinking Targets versus Recurrence

Let (X,d) be a compact metric space and (X,𝒜,μT) be a probability measure-preserving system. Furthermore, given a real, positive function ψ : â„• → â„, let

R(ψ) := { x ∈ X : d(Tⁿx, x) < ψ(n) for infinitely many n ∈ â„• }

denote the associated recurrent set, and given a point y ∈ X, let

W(ψ) := { x ∈ X : d(Tⁿx, y) < ψ(n) for infinitely many n ∈ â„• }

denote the associated shrinking target set. Under certain mixing properties, it is known that if ∑ψ(n) diverges, then both the recurrent and shrinking target sets are of full μ-measure. The purpose of this talk is to discuss the potential quantitative strengthening of these full measure statements.